See for example [1]}, where \(\nabla ^s\)  is denoted as \(\nabla ^b\)  (note that our \(T^j_{ik}\)  here is twice as much as that in there). As a consequence, we get (see (22) in [1]}) the components of \(T^s\)  as
\(T^s(e_i,e_j) = - \sum _k T^k_{ij}e_k, \ \ \ \ T^s(e_i, \overline{e}_j) = \sum _k \big ( T^j_{ik} \overline{e}_k - \overline{T^i_{jk}} \,e_k \big )\) 
